As \(I_T(\phi )=\infty \)  if \(\phi \)  is not absolutely continuous, we can restrict our attention to sequences of absolutely continuous functions. As the Skorohod topology relativized to \(C([0,T];A)\)  coincides with the uniform topology (see, e.g., [1]}, Section 12, p.124), we can consider a sequence of functions \(\phi _n \in C([0,T];A)\)  converging to a function \(\phi \)  under the uniform topology. We can furthermore assume that \(I_T(\phi _n) \le K\)  for some \(K\)  and all \(n\ge 1\) .
By Lemma REF , the functions \(\phi _n\)  are hence uniformly absolutely continuous and therefore the limit \(\phi \)  is absolutely continuous.
